Fall 2010 McNabb GDCTM Contest
Algebra II
NO Calculators Allowed
1. An automobile goes y/9 yards in d seconds. How many feet does it travel in two
minutes time?
40y 40d 3y 120y
(A) (B) (C) (D) 120yd (E)
d 3y 40d d
2. The well-known formula f = (9/5)c + 32 relates the temperature f in Fahrenheit
to the temperature c in Celcius. For how many values of f satisfying 32 ≤ f ≤
212, will the temperature be an integer in both of these scales?
(A) 9 (B) 10 (C) 19 (D) 20 (E) 21
√ x+3
3. If f ( x ) = and g( x2 ) = x4 + 3x2 − 22, then find f ( g(4)).
13
(A) 3 (B) 4 (C) 5 (D) 6 (E) 7
4. A square is inscribed in a right triangle with sides of length 3, 4, and 5, so that
one of the sides of the square is contained in the hypotenuse of the right triangle.
What is the side length of the square?
60 12
(A) (B) 2 (C) (D) 3 (E) cannot be determined
37 5
5. The arithmetic mean of a, b, and c is 7 and the arithmetic mean of a2 , b2 , and c2
is 55. What is the arithmetic mean of ab, bc, and ac?
(A) 24 (B) 31 (C) 46 (D) 48 (E) 92
6. In how many ways can the the letters in the string ABECEDA be arranged so
that the consonants are in alphabetical order?
(A) 90 (B) 105 (C) 120 (D) 180 (E) 210
Fall 2010 Algebra II 1
7. The graph of the quadratic function f ( x ) = ax2 + bx + c contains the points
(−1, 6), (7, 6), and (1, −6). What is the minimum value of f ( x )?
(A) -36 (B) -26 (C) -20 (D) -10 (E) -6
8. Let a, b, and c be positive real numbers. Supposing that ab = kc, ac = lb, and
bc = ma, then c must equal
√ √ √
r
l
(A) lm (B) klm (C) (D) k lm (E) lm
m
p √ √ √
9. The real number 16 + 220 can be expressed in the form A + B, where A
and B are integers and A > B. What is the value of A − B?
(A) 6 (B) 7 (C) 8 (D) 9 (E) 10
D C
I
III
II
10. In trapezoid ABCD, the area of region I is
A
9 and the area of region II is 16. What is B
the area of region III?
(A) 10 (B) 11 (C) 12 (D) 12.5 (E) cannot be determined
11. The polynomial
x8 + x7 + x6 + x5 + x4 + x3 + x2 + x + 1
can be factored as P2 ( x ) · P6 ( x ) where P2 and P6 are polynomials with integer co-
efficients and have degrees 2 and 6 respectively. Find the sum of the coefficients
of P2 .
(A) 1 (B) 2 (C) 3 (D) 6 (E) 9
Fall 2010 Algebra II 2
B
12. In 4 ABC, ∠ A = 60◦ , C = 40◦ , BD ⊥ AC,
−
→
and BE bisects ∠ ABC. Find the measure
A D E C
of ∠ DBE in degrees.
(A) 8 (B) 10 (C) 12 (D) 14 (E) 20
13. A bag contains 4 quarters and 2 dimes. If 3 coins are randomly removed from
the bag, what is the expected total value in cents of these three coins?
(A) 50 (B) 55 (C) 60 (D) 65 (E) 75
14. The set S contains seven numbers whose mean is 202. The mean of the four
smallest numbers in S equals 100, while the mean of the four largest numbers in
S equals 300. What is the median of all the numbers in S?
(A) 184 (B) 186 (C) 192 (D) 196 (E) 200
15. Let a and b be positive constants. If x is a solution of
√ √ √
x+a+ x+b = a+b
then x must equal
a+b ab
(A) 0 (B) (C) −
2 a+b
1 1 2
(D) − + (E) −
a b a+b
16. While Xerxes marched on Greece his army streched out for 50 miles. A dispatch
rider had to ride from the rear to the head of the army, then instantly turn about
and return to the rear. While he did this, the army advanced 50 miles. How
many miles did the rider ride?
√ √
(A) 100 (B) 50 + 50 √2 (C) 100 2
(D) 150 (E) 50 + 100 2
Fall 2010 Algebra II 3
17. The sum of two of the roots of p( x ) = 4x3 + 8x2 − 9x − k, where k is a constant,
is zero. Find the value of k.
(A) 3 (B) 6 (C) 12 (D) 18 (E) 200
18. Let f be a function such that f ( x + y) = f ( xy) for all real numbers x and y. If it
is also known that f (5) = 5, determine the value of f (25).
(A) 1 (B) 5 (C) 10 (D) 20 (E) 25
19. Hezy and Zeke were employed at different daily wages. At the end of a certain
number of days Hezy received $300, while Zeke, who had been absent from
work two of those days, received only $192. However, had it been the other way
around, had Zeke worked all those days and Hezy been absent twice, then both
would have received the same amount. What was Hezy’s daily wage?
(A) 30 (B) 40 (C) 50 (D) 60 (E) 70
20. A five-digit integer, with all distinct digits which in this problem must be 1,2,3,4,
and 5 in some order, is called alternating if the digits alternate between increasing
and decreasing in size as read from left to right. They may start on an increas-
ing or decreasing foot. For instance, both 34152 and 53412 are alternating while
12354 is not, for example. How many of this kind of 5 digit integer are alternat-
ing?
(A) 32 (B) 28 (C) 24 (D) 20 (E) 16
Fall 2010 Algebra II 4
